Bent functions with stronger nonlinear properties: K-bent functions

Abstract

We introduce the notion of k-bent function, i.e., a Boolean functionwith evennumber m of variables υ1, …, υm which can be approximated with all functions of the form 〈u, v〉j ⊕ a in the equally poor manner, where u ∈ ℤ2m, a ∈ ℤ2, and 1 ⩽ j ⩽ k. Here 〈·, ·〉j is an analog of the inner product of vectors; k changes from 1 to m/2. The operations 〈·, ·〉k, 1 ⩽ k ⩽ m/2, are defined by using the special series of binary Hadamard-like codes Amk of length 2m. Namely, the vectors of values for the functions 〈u, v〉k ⊕ a are codewords of the code Amk. The codes Amk are constructed using subcodes of the ℤ4-linear Hadamard-like codes of length 2m+1, which were classified by D. S. Krotov (2001). At that the code Am1 is linear and Am1, …, Amm/2 are pairwise nonequivalent. On the codewords of a code Amk, we define a group operation • coordinated with the Hamming metric. That is why we can consider k-bent functions as functions that are maximal nonlinear in k distinct senses of linearity at the same time. Bent functions in usual sense coincide with 1-bent functions. For k > l ⩾ 1, the class of k-bent functions is a proper subclass of the class of l-bent functions. In the paper, we give methods for constructing k-bent functions and study their properties. It is shown that there exist k-bent functions with an arbitrary algebraic degree d, where 2 ⩽ d ⩽ max {2, m/2 − k + 1}. Given an arbitrary k, we define the subset \( \mathfrak{F}_m^k \mathcal{F}_m^k \) of the set \( \mathfrak{F}_m^k \mathcal{F}_m^k \)\( \mathcal{A}_m^k \mathcal{B}_m^k \) of all Boolean functions in m variables with the following property: on this subset k-bent functions and 1-bent functions coincide.